Adding some mesh to existing rigged mesh
There is a data transfer modifier, which can transfer the weights between meshes. Depending on the meshes you might need to go in and change some things manually afterwards, but it sure takes care of most of it.
https://docs.blender.org/manual/en/latest/modeling/modifiers/modify/data_transfer.html
I would go with the Nearest Vertex sampling method. Topology option would not make sense in your case as the topologies are different.
The meshes should be joined after the data transfer modifier is added and applied